BISHOPS
GEOMETRY Grade 9
Revision of Parallel Lines, Angles, Triangles and Proofs
2022
LINE AND ANGLES
THEOREM STATEMENT
ACCEPTABLE REASON(S)
Two adjacent angles on a
straight line are supplementary
∠s on a str line
The sum of two or more angles
on a straight line is 180°
(x + y + z = 180°)
∠s on a str line
If the adjacent angles are
supplementary, the outer arms
of these angles form a straight
line.
adj ∠s supp
The angles in a revolution add
up to 360°.
∠s round a pt
(x + y + z = 360°)
∠s in a rev
Vertically opposite angles are
equal.
vert opp ∠s =
If AB || CD, then the alternate
angles are equal.
alt ∠s; AB || CD
If AB || CD, then the
corresponding angles are equal.
corresp ∠s; AB || CD
OR
(a and b are vertically opp ∠s)
(c and d are vertically opp ∠s)
(e and f are alternate ∠s)
(e and g are corresponding ∠s )
If AB || CD, then the co-interior co-int ∠s; AB || CD
angles are supplementary.
( f and h are co-interior ∠s)
(f + h = 180°)
If the alternate angles between
two lines are equal, then the
lines are parallel.
alt ∠s =
(You can prove that lines PQ and
RT are parallel if you can show
that the alternate angles are equal)
1
DIAGRAM
If the corresponding angles
between two lines are equal,
then the lines are parallel.
corresp ∠s =
If the cointerior angles
between two lines are
supplementary, then the lines
are parallel.
coint ∠s sup
(You can prove that lines RT and
QP are parallel if you can show
that the corresponding angles are
equal)
(You can prove that lines RT and
QP are parallel if you can show
that the co-interior angles are
supplementary)
TRIANGLES
THEOREM STATEMENT
ACCEPTABLE REASON(S)
The interior angles of a triangle are
supplementary.
∠ sum in ∆
OR
sum of ∠s in ∆
OR
Int ∠s ∆
The exterior angle of a triangle is
equal to the sum of the interior
opposite angles.
ext ∠ of ∆
The angles opposite the equal sides
in an isosceles triangle are equal.
∠s opp equal sides
(w=x+y)
(You can prove that x = y )
(You may NOT say ISOSCOLES
∆ as the reason)
The sides opposite the equal angles
in an isosceles triangle are equal.
sides opp equal ∠s
(You can prove that AB = AC)
(You may NOT say ISOSCOLES
∆ as the reason)
2
DIAGRAMS
Revision of Grade 8 Geometry
Study the examples below and answer the questions that follow:
Worked example:
°
ˆ
In the figure below
=
=
QPR
105
and Rˆ 35° . Find, in order, the sizes of angles a to f.
a = 140
exterior ∠ of ∆ PQR
b = 140
corresponding ∠ s, ST || MR
c = 40
<’s on straight line or co-interior ∠ s, ST || MR
d = 40
vertically opposite ∠ s
e = 35
alternate ∠ s, ST || MR
f = 40
<’s on straight line or alternate ∠ s, ST || MR or sum <’s in ∆ PQR
Solve for x by setting up an equation and then solving it. Give appropriate reasons.
Worked example
x + 3x + 10° +=
x − 20° 180°
°
°
sum < 's in ∆KLM
°
5 x = 180 − 10 + 20
5 x = 190°
x = 38°
Worked example
2 x =x + 48°
ext ∠ of ∆EFG
2x − x =
48°
x = 48°
3
Exercise 1
ˆA=
1. In the figure below, FD|| BC and FE=BE . B Aˆ C =
52° and B C
20°
Write down the size of each angle labelled a to f (in that order).
a = _____________________________________________________
b = ____________________________________________________
c = ____________________________________________________
d = ____________________________________________________
e = _____________________________________________________
f = _____________________________________________________
2.
ˆ = 2 x + 5° and A Cˆ B = 70°
In the diagram AB|| DE . A
2.1
Write down the value of B̂ in terms of x
2.2
Set up an equation and solve for x
2.3
Find the size of BCˆ E
3.
Set up an equation and solve for x.
4
Writing angles in term of x and y
Fill in the size of each of the missing angles in terms of x or y.
(Although no working needs to be shown, you may still show your calculations if you find
it helps you to find the values of the missing angles.)
Worked example
Optional Calculation:
B̂ = 180 − ( 90 + x )
= 90 − x
Worked example
Optional calculations:
F̂ = x
D̂= 180 − ( x + x )
= 180 − 2 x
Worked example
Optional calculations:
180 − 2 x
ˆ
ˆ
K= L=
2

= 90 − x
Worked example
5
Exercise 2
1. Fill in on the diagram the size of each of the missing angles in terms of x or y.
(Although no working needs to be shown, you may show your calculations if you
prefer.)
1.1
1.2
1.3
1.4
1.5
1.6
1.7
6
Geometric Proofs
If A = B and A = C
This logic is used often in geometric proofs
then B = C
and is called deductive reasoning.
Worked Example:
P
Q
C
N 1
1
D
A
B
E
S
If CÂB = CB̂A,
R
1
M
2
K
T
If N1 = K1
prove that CÂD = CB̂E
Prove QR//ST
PROOF: Let N=
K=
x
1
1
PROOF : Let CÂB = x
(given )
CÂD = 180 ° − x (∠' s on str lineDE )
∴ CB̂A = x
CB̂E = 180 ° − x (∠' s on str line DE)
M=
K=
x < ' s opp equal sides
1
1
∴ N 1 = M1 = x
∴ QR//ST
corres∠'s =
∴ CÂD = CB̂E
Worked Example:
A
Prove : AC = CD
Proof:
ˆ
ˆ
BAC
=
ACB
=
68 exterior angle of isosΔ
ˆ = 34 exterior angle of isosΔ
ADC
34°
E
136°
B
C
ˆ = ADC
ˆ = 34
∴ CAD
D ∴ AC= CD sides opp= angles ∴Δis isos
7
Exercise 3. Remember to give reasons and follow the deductive reasoning path.
P
A
1.
2.
1
L
Q
M
1
1
Q
2
S
R
N
1
1
P
1
B
C
Prove that M 1 + P1 = 180 °
Prove that Q̂1 = B̂1
3.
A
C
B
1
2
C
4.
B
1
2
D
A
E
1
Given: B̂1 = B̂ 2
Given: Â 1 = D̂ 2
Prove: D̂ = Ê
Prove:
8
B̂1 = D̂1 + D̂ 2
1
2
D
B
A
5·
D
1
6·
1
E
C
B
A
140°
D
100°
C
E 120°
F
Given: EĈB = 100 0
Prove: AB // EF
Given: AB = AC
Prove : D̂1 = Ê 1
B
B
7.
8·
A
E
C
1
A
P 2
2
D
1
D
Prove :
 1 = D̂1
3
B
Given: B̂1 = B̂ 2
Prove: Â = Ĉ
9
C
and R̂ 1 = R̂ 2
9. Given: Q̂1 = Q̂ 2
Prove:
T̂ = 90
T
1
Q
10.
P
2
Given: P̂ =R̂ 2
Prove: PS = SQ (hint use isosceles triangles)
x
S 2
1
x
Q
1
2
R
Given : EC = BC
Prove : BE bisects AB̂D
11.
A
E
D
B
1 2 R
C
10
12.1 Find, with reasons, the value of x
(
12.2 Prove that DE||AC
13.
In the diagram AB|| EC and EC bisects A Cˆ D
Prove that AC = BC. Hint: Let A Cˆ E = x
14.
In the diagram, all the lines are straight lines and intersect at O
Prove that C G ⊥ A E
11
15.
16.
In the diagram TP = TA, N E ⊥ P A and P̂ = x
ˆ = ANE
ˆ
Prove that PUE
In the diagram AB = AC and BC = CD .
ˆ . Hint: Let BDC
ˆ =x
Prove that BCˆ D = BAC
17.
In the diagram, PATE is a square with P̂1 = Aˆ 1
Prove that P G ⊥ A U
Hint: Let P̂1 = x
12
[4]
PYTHAGORAS
1.
The diagram shows a square of area x square units
inscribed inside a semicircle and a larger square of
area y square units inscribed inside a circle.
What is the ratio x:y ?
2.
What is the area of this quadrilateral, in cm2?
3.
A 15m ladder leans against a wall. If it is 5m away from the foot of the wall:
4.
(i)
Determine how far up the wall the ladder reaches
(leave your answer in simplest surd form).
(ii)
The ladder slips 2 2 m down the wall. How far away from the wall is the
ladder now?
A light pole that is 15m high bends in the wind at a fixed point and falls to the
ground in a way that it is divided in the ratio 1:2.
(i)
What is the shorter length of the bent light pole?
(ii)
How far away does the bent light pole land from its original
position?
13
5.
Find the length of a diagonal of a rectangle whose base is 14 cm and whose height is
5 cm.
6.
A window is 8 m above ground level. There is a 2 m wide flower bed at the base of
the wall which must not be disturbed, How long a ladder would you need to be able
to reach the window?
7.
Four unit squares are placed edge to edge as shown.
What is the length of the diagonal line drawn?
8.
A tunnel has a semi-circular cross section and a
diameter of 10 m. If the roof of a bus just touches the
roof of the tunnel when its wheels are 2 m from one
side, the height, including the wheels (in metres) of
the bus is…
9.
Refer to the diagram alongside. The length of one side
of the large square is 4 cm and the length of one side
of the small square is 3 cm.
Find the area of the shaded region in cm2.
10.
Find the area of the following triangles below.
14
Find the area of ∆ABC and the value of x.
Find the area of ABCD.
Find the area of the quadrilateral.
Find the area of the quadrilateral.
Find AB if the area of the triangle is 72cm2. Find x if the area of the triangle is 1638cm2.
15
EC = 6 cm, EB = 4 cm and CB = 8 cm.
The area of ∆ECB = 48 cm2.
Find the lengths of AB, CD and EF.
11.
12.
Determine if the following triples are Pythagorean.
11.1 3, 4, 5
11.4 7, 24, 25
11.2 6, 8, 10
11.5 10, 25, 26
11.3 5, 12, 13
11.6 9, 13, 15
Look at the following triples:
3, 4, 5
and
5, 12, 13
and
7, 24, 25
• Note that the smallest number in the triple is an odd number.
• Note that the difference between the other numbers is 1.
See if you can find triples in which the smallest number is
12.1 9
12.2 11
12.3 13
13.
14.
16